Magnetic Attributes of NiFe₂O₄ Nanoparticles: Influence of Dysprosium Ions (Dy³⁺) Substitution

Abstract

This paper reports the influence of dysprosium ion (Dy³⁺) substitution on the structural and magnetic properties of NiDy_xFe_2−xO₄ (0.0 ≤ x ≤ 0.1) nanoparticles (NPs) prepared using a hydrothermal method. The structure and morphology of the as-synthesized NPs were characterized via X-ray diffraction (XRD), scanning and transmission electron microscope (SEM, and TEM) analyses. ⁵⁷Fe Mössbauer spectra were recorded to determine the Dy³⁺ content dependent variation in the line width, isomer shift, quadrupole splitting, and hyperfine magnetic fields. Furthermore, the magnetic properties of the prepared NPs were also investigated by zero-field cooled (ZFC) and field cooled (FC) magnetizations and AC susceptibility measurements. The M_ZFC (T) results showed a blocking temperature (T_B). Below T_B, the products behave as ferromagnetic (FM) and act superparamagnetic (SPM) above T_B. The M_FC (T) curves indicated the existence of super-spin glass (SSG) behavior below T_s (spin-glass freezing temperature). The AC susceptibility measurements confirmed the existence of the two transition temperatures (i.e., T_B and T_s). Numerous models, e.g., Neel–Arrhenius (N–A), Vogel–Fulcher (V–F), and critical slowing down (CSD), were used to investigate the dynamics of the systems. It was found that the Dy substitution enhanced the magnetic interactions.

1. Introduction

Lately, nanosized Ni-spinel ferrites have widely been used in magnetic storage media, magnetic adsorbents, telecommunication, catalysts, microwave absorbers, and computer memories [1,2,3,4,5]. Further, the structural and magnetic characteristics of nanosized Ni-spinel ferrites have been enhanced by substituting varieties of magnetic, nonmagnetic, and rare earths ions [6,7,8]. Such improvement was majorly attributed to the distribution of substitution ions between tetrahedral and octahedral sites of the host and eventual influence on their magnetic moments [8]. Moreover, the use of rare earths as substitution agents in the spinel ferrite was found to cause structural distortion through symmetry preserving the lattice strain and thereby modifying the overall properties of nanosized spinel ferrites [9]. Intense research efforts have been made to enhance the structure, optical, and magnetic properties of rare earth ion substituted nanosized Ni-spinel ferrites [2,3]. Meanwhile, different theories and models have been applied to achieve a better understanding of the unique magnetic properties of these doped nanosized Ni-spinel ferrites [2,3]. Modified magnetic properties of Erbium (Er³⁺) and Samarium (Sm³⁺) substituted nickel ferrites were analyzed by low temperature magnetization, and zero-field cooled and field cooled (ZFC-FC) measurements. Such samples revealed a superparamagnetic (SPM) behavior with very low coercivity (H_c) and remanence magnetization (M_r), making them suitable for developing soft magnets [10]. A substitution of Pr, Sm, and La in Ni_0.5Zn_0.5Fe_1.95R_0.05O₄ nanoparticles (NPs) was found to modify their magnetic and dielectric properties [11]. Conversely, Cerium (Ce) substituted Ni-Zn at low contents (0.0 ≤ x ≤ 0.1) produced high saturation magnetization (M_s), remanence, and coercivity [12].

So far, studies on Dy³⁺ substituted nanosized Ni spinel ferrites are rare. J. Sahariya et al. [13] reported the temperature dependent spin momentum densities of NiFe_2−xRE_xO₄ (x = 0, 0.05 and RE = Dy, Gd) ferrites measured by using a magnetic Compton spectrometer. The experimental profiles of NiFe₂O₄ with doping of Dy and Gd show a similar spin moment to the non-doped sample. The contribution of different constituents in the formation of total spin moment is deduced from the analysis of the Compton line shape. It is seen that Dy³⁺ or Gd³⁺ doping ions at Fe³⁺ sites lead to a redistribution of the spin moment at Fe³⁺ and RE³⁺ sites. The temperature dependent magnetic Compton profiles are decomposed into the constituent profiles of Ni, Fe, Dy/Gd, and diffuse components, arising due to O-2sp states. A decrease in the Fe spin moment from 0.55 ± 0.03 μ_B/f.u. (in NiFe₂O₄) to 0.41(0.50) ± 0.03 μ_B/f.u. is observed on the partial substitution of Dy(Gd). K. Kamala Bharathi et al. [14] reported a correlation between the microstructure, electrical, and optical properties of Dysprosium-doped nickel-ferrite (NiFe_1.925Dy_0.075O₄) thin films fabricated using sputter-deposition using a stoichiometric bulk target prepared by tsolid-state chemical reaction. Recently, M.A. Almessiere et al. [15] investigated the effects of dysprosium substitution on the structural, microstructural, and magnetic properties of NiFe₂O₄ nanoparticles. The formation of the cubic phase of Ni nanosized ferrites was confirmed. The magnetic properties were done by analyzing measurements of the magnetization versus applied field M(H). These measurements were carried out at two different temperatures—room temperature (T = 300 K) and low temperature (T = 10 K). A noticeable improvement in the differently deduced magnetic parameters, including saturation magnetization (M_s), remanence (M_r), and coercivity (H_c), was observed at both room (T = 300 K) and low (T = 10 K) temperatures with Dy substitution. The increase in the different magnetic parameters is mainly attributed to the strengthening of A–B exchange interactions owing to the substitution of Fe³⁺ ions with Dy³⁺ ions with larger ionic radii, the formation of local strains, and the increase in the magnetic moments (N_B).

To the best of our knowledge, there is no study reporting the Mossbauer, zero-field cooled (ZFC), and field cooled (FC) magnetizations and AC susceptibility measurements for Dy substituted nanoparticles of Ni spinel ferrites. Accordingly, we examined in the present study the influence of Dy³⁺ doping on the structure, morphology, Mossbauer, ZFC-FC magnetization, and AC susceptibility of NiDy_xFe_2-xO₄ (0.0 ≤ x ≤ 0.1) nanoparticles. Various products were synthesized using a hydrothermal process.

2. Experimental

Various specimens with the chemical formula NiDy_xFe_2−xO₄ (0.0 ≤ x ≤ 0.1) have been prepared by a hydrothermal process. Analytical grade high purity chemical reagents including Dysprosium III nitrate hydrate (Dy (NO₃)₃ H₂O), Nickel nitrate (Ni(NO₃)₂ 6H₂O), Iron III nitrate (Fe(NO₃)₃ 9H₂O), and Sodium Hydroxide (NaOH) were taken as starting materials. Then, the stoichiometric amount of these compositions was mixed in distilled water and the resulting mixture was left at room temperature. Next, a clear solution was made for pH adjustment using 2M of NaOH solution and placed in a stainless steel-Teflon autoclave to initiate the hydrothermal treatment. Finally, the obtained sample was dried and grinded for further experimental analyses.

A Rigaku Benchtop Miniflex X-ray diffraction (XRD) diffractometer (Tokyo, Japan) with Cu K_α radiation at room temperature (RT) over a 2θ range from 20° to 70° was used for the structural analysis. Scanning electron microscopy (SEM, FEI Titan 80–300 kV FEG S/TEM, Hillsboro, OR, USA), along with energy dispersive X-ray spectroscopy (EDX) and transmission electron microscopy (TEM; FEI, Morgagni 268, Prague, Czech Republic), was used for the morphological and composition analyses. AC magnetic susceptibility, zero-field cooled (ZFC), and field cooled (FC) magnetization measurements were performed using a superconducting quantum interference device (PPMS DynaCool, Quantum Design, San Diego, CA, USA). ZFC-FC magnetizations were carried out at a temperature range of 2–400 K under a DC field of 100 Oe. Real and imaginary parts of AC susceptibility measurements were done at temperatures ranging from 350 to 2 K in the presence of an ac applied magnetic field of $H_{\mathrm{ac}}$ = 10 Oe and in different frequencies ranging from 50 to 10,000 Hz. Mössbauer spectra were performed at room temperature using a conventional Mössbauer spectrometer (Fast Com Tec PC-moss II, Oberhaching, Germany) under the constant acceleration mode using ⁵⁷Fe in an Rh matrix with an approximate activity of 10 m Ci. The speed scale and the velocity were calibrated using α-Fe and laser interferometry, respectively. The recorded spectra were analyzed and fit to the inbuilt Win-Normos fitting software (WISSEL company, Duisburg, Germany).

3. Results and Discussion

3.1. Structure

Figure 1 displays the XRD powder patterns of the studied NiDy_xFe_2−xO₄ (0.0 ≤ x ≤ 0.1) nanoparticles. Irrespective of Dy³⁺ contents, all the XRD peaks have been verified to the cubic nanosized Ni-spinel ferrites (JCPDS Card Number 54-0964), indicating the lattice site compatibility of the Dy³⁺ ions in the Ni spinel ferrite structure in the absence of any impurity. The calculated structural parameters of the prepared nanosized spinel ferrites were enlists in

Table 1. Structural parameters of studied NiDy_xFe_2−xO₄ NPs.

x	a (Å)	V (Å3)	DXRD (nm)
0.00	8.3378	579.6451	34.7
0.01	8.3397	580.0333	33.1
0.03	8.3413	580.3630	34.7
0.05	8.3424	580.3463	30.8
0.07	8.3473	579.3824	32.3
0.09	8.3470	579.9142	33.1
0.10	8.3473	580.9998	24.3

. It was realized that lattice parameters elongate due to the increase in the substitution contents and also show ionic radii disparity between Fe³⁺ (0.78 Å) and Dy³⁺ (1.03 Å). The calculated crystallite sizes (D_XRD) were in the range of 24 to 35 nm [16].

3.2. Morphology

Figure 2 displays the SEM pictures, EDX spectra, and elemental mapping results for x = 0.01 and 0.05 samples. The surface morphology of these samples revealed a high degree of nanoparticle agglomeration. EDX analysis proved the existence of appropriate elements (Fe, Ni, Dy, and O) in the samples. Figure 3 illustrates the TEM image of the nanosized spinel ferrites containing Dy³⁺ contents of 0.05. The inset (Figure 3) displays the selected angle electron diffraction (SAED) patterns of the corresponding sample that confirmed the agglomeration of single crystalline Ni spinel ferrite nanoparticles.

3.3. Mössbauer Spectra

Figure 4 depicts the Mössbauer spectra of synthesized NiDy_xFe_2−xO₄ (0.0 ≤ x ≤ 0.1) NPs at room temperature.

Table 2. Evaluated Mössbauer parameters of studied ferrite NPs (B_hf: hyperfine magnetic field, I.S.: isomer shift, Q.S.: quadrupole splitting, W: line width, and R_A: Relative area).

x	Assignment of Sites	I.S. (±0.002) (mm/s)	Q.S. (±0.004) (mm/s)	Bhf (±0.02) (T)	W (±0.01) (mm/s)	RA (%)	Cation Distribution
0.01	Sx-A: Fe3+	0.368	0.005	52.045	0.218	17.298	(Ni0.65Fe0.35)A [Ni0.35 Dy0.01Fe1.64]BO4
	Sx-B: Fe3+	0.438	0.097	49.407	0.401	35.575
	Sx-B1: Fe3+	0.202	0.011	48.566	0.37	46.375
	Db: Fe+3+	0.226	0.717	−	0.399	0.752
0.03	Sx-A: Fe3+	0.364	0	52.009	0.205	17.358	(Ni0.66Fe0.34)A [Ni0.34Dy0.03Fe1.63]BO4
	Sx-B: Fe3+	0.455	0.119	49.337	0.383	34.389
	Sx-B1: Fe3+	0.204	0.09	48.55	0.38	48.253
0.05	Sx-A: Fe3+	0.365	0.008	51.941	0.257	16.783	(Ni0.67Fe0.33)A [Ni0.33Dy0.05Fe1.62]BO4
	Sx-B: Fe3+	0.438	0.071	49.379	0.395	35.528
	Sx-B1: Fe3+	0.231	−0.018	48.559	0.402	47.689
0.07	Sx-A: Fe3+	0.366	−0.008	52.057	0.242	17.866	(Ni0.65Fe0.35)A [Ni0.35Dy0.07Fe1.58]BO4
	Sx-B: Fe3+	0.464	0.072	49.737	0.466	25.675
	Sx-B1: Fe3+	0.218	−0.023	48.76	0.422	54.952
	Db: Fe3+	0.345	0.625	−	0.725	1.5067
0.09	Sx-A: Fe3+	0.365	0.004	52.119	0.219	19.981	(Ni0.62Fe0.38)A [Ni0.38Dy0.09Fe1.53]BO4
	Sx-B: Fe3+	0.424	0.066	49.614	0.445	35.589
	Sx-B1: Fe3+	0.209	−0.005	48.766	0.394	45.43
0.10	Sx-A: Fe3+	0.37	−0.004	52.123	0.239	20.331	(Ni0.6Fe0.4)A [Ni0.4Dy0.1Fe1.5]BO4
	Sx-B: Fe3+	0.441	0.053	49.544	0.397	31.53
	Sx-B1: Fe3+	0.198	−0.02	48.825	0.413	46.22
	Db: Fe3+	0.379	0.481	−	0.696	1.9186

enlists various Mössbauer parameters calculated by spectral fitting using three sextets (A for the tetrahedral sites and B and B₁ for the octahedral sites). Fe³⁺ ions in the tetrahedral A site are characterized by a large hyperfine field with an insignificant isomer shift. Conversely, the other two sextets with a comparatively smaller hyperfine field signify the occupation of Fe³⁺ at two dissimilar environments in the B-site [17]. Besides the ferromagnetic sextets, a minute paramagnetic doublet with quadrupole-splitting was evidenced for NiDy_0.01Fe_1.99O₄, NiDy_0.07Fe_1.93O₄, and NiDy_0.1Fe_1.9O₄ NPs. The occurrence of such a paramagnetic doublet was attributed to the fractions of Fe³⁺ with fewer nearest neighbors that possessed magnetically ordered spins. Interestingly, in the spinel ferrite structure, Fe³⁺ did not contribute to the super exchange interaction [18].

The achieved relative area for the A and B sites clearly indicated the occupation of Ni³⁺ in the A and B sites. Kumar et al. acknowledged the preferential occupation of Dy³⁺ in the octahedral B sites of Co-ferrites [19,20]. Thus, the cation distribution in the proposed Dy³⁺ substituted nanoferrites was obtained following the formula unit of (Ni_yFe_1−y)_A (Ni_1−yDy_xFe_1+y−x)_B. The distribution of Fe³⁺ over the A and B sites was observed to be relative to the proportional area of A and B in the Mossbauer sub-spectra. Table 2 summarizes the approximate cation distribution obtained from the Mössbauer spectra. The results in Table 2 revealed that Fe³⁺ cations emigrated from the B site to A site due to Dy³⁺ substitution. The line width of the A site was randomly altered, whereas for B and B₁ sites, it was enhanced with the substitution of Fe³⁺ (0.64 Å) in the B sites by Dy³⁺ (0.91Å) having larger ionic radii than the former one. This observation authenticated the increase in the degree of disorder due to the substitution on B sites.

The values of the hyperfine magnetic fields for the A and B sites (Table 2) in the studied nanosized spinel ferrites were first reduced with an increase in Dy³⁺ contents up to 0.05 and then enhanced at 0.07. Eventually, the hyperfine field for A site was continuously enhanced, but for the B site it was diminished. This alteration in the hyperfine field for the A and B sites was attributed to the addition of the diamagnetic Dy³⁺ that replaced the ferromagnetic Fe³⁺ with a higher magnetic moment (5 µ_B) and lowered the average number of magnetic linkages (${\mathrm{Fe}}_{\mathrm{A}}^{3+}-\mathrm{O}-{\mathrm{Fe}}_{\mathrm{B}}^{3+}$). Thus, Fe³⁺ nuclei experienced a reduction in the magnetic field at both the sublattices up to the Dy3+ content of 0.05. Beyond 0.05, the number of Fe³⁺ at the A site was augmented, thereby increasing the hyperfine magnetic field and magnetic moment of Fe³⁺ at the A site.

3.4. ZFC-FC Magnetizations

Figure 5 shows the curves of zero-field-cooled (ZFC) and field-cooled (FC) temperature dependencies of the magnetization, M_ZFC (T) and M_FC (T), of NiFe_2−xDy_xO₄ (where x = 0.00, 0.03 and 0.09) NPs. These measurements were performed in a temperature interval ranging between 2 and 400 K under a DC field of 100 Oe. For M_ZFC (T) measurements, the sample was cooled, first of all, from room temperature (RT) to a very low temperature in the absence of an applied field and subsequently the magnetization was recorded by increasing the temperature in the presence of the field. However, in the M_FC (T) measurements, the magnetization was recorded by cooling the product in the presence of applied field. A splitting and a large irreversibility between M_ZF (T) and M_FC (T) curves for different synthesized products can be clearly seen in Figure 5. The M_FC (T) increased gradually and remained constant below temperature $T_{\mathrm{s}}$, while the M_ZFC (T) decreased with a lowering of the temperature down to about 4 K. The dispersion of M_ZFC-M_FC versus T curves is congruent with the poly-disperse character of magnetic NPs, with a correlated distribution in particle size and individual anisotropy axes [21]. The enlargement could also owe to dipolar interactions among particles [21].

It is reported in the literature that the manifestation of a peak in the M_ZFC (T) plots is associated to the blocking temperature (T_B) [22]. The curves of M_ZFC (T) of the prepared products showed an incomplete maximum or broad maximum at around the temperature noted T_B. This is typical for superparamagnet (SPM) materials, which show the properties of classical paramagnet materials (PM) above T_B, where the total spin is equal to the spin of a whole NPs but behave as ferromagnetic (FM) materials below their blocking temperature (T_B). Below T_B, the M_FC (T) and M_ZFC (T) curves considerably diverge, and the various ferrite NPs are in the FM state (blocked state). Above T_B, the M_FC (T) and M_ZFC (T) curves coincide, which is because all NPs are in the same SPM state. At T = T_B, the thermal activation overcomes the magnetic anisotropy barrier, which leads to fluctuations in magnetization [23]. Therefore, the wide peak at T_B in the M_ZFC (T) curves is an indication of a broadened energy barrier distribution. Further, it can be seen that the blocking temperature varies by increasing Dy substitution content. The non-substituted product NiFe₂O₄ shows a blocking temperature around T_B ≈ 390 K. The x = 0.03 product exhibits a well-defined T_B at around 300 K. By further increasing the amount of Dy, the product synthesized with x = 0.09 was not able to reach the T_B within 400 K, so the T_B value is superior than 400 K for x = 0.09. It can be seen clearly that the blocking temperature decreases for a lower Dy content (x = 0.03) and then increases for higher content (x = 0.09). The dependence of the T_B on particles size has been reported in previous studies [24]. The lower T_B is attributed to smaller particle size or narrow size distribution. However, the higher T_B represents a larger particle size. Nevertheless, the different x = 0.00, 0.03 and 0.09 products show approximately same particles size. Therefore, the variations in blocking temperature with substitution effects are not predominantly influenced by the grain size. Thus, in addition to the particle size effect, the T_B could also be affected by numerous other extrinsic factors, mostly related to interactions among particles and intrinsic factors that principally include a magneto–crystalline, surface and shape anisotropy [22,25]. We noticed in the synthesized product with x = 0.03 that the M_ZFC (T) exhibits a breaking at temperatures indicated by the dashed circle in Figure 5.

On the other hand, the M_FC (T) curves increase smoothly for different samples with a decrease of temperature, while a kind of saturation in the magnetization is noticed below the temperature noted by $T_{\mathrm{s}}$ for all samples. It is reported in the literature that for SPM nanoparticles, the curve of M_FC (T) increases continuously [26,27]. Nevertheless, in the case of super-spin glass (SSG) systems in which the interactions among particles are strong, a flat type or a slow increase is observed [26,27]. Therefore, the detected flat nature below $T_{\mathrm{s}}$ in the M_FC (T) curves establishes the occurrence of an SSG-like state. The origin of the observed magnetic features in ZFC-FC magnetization part will be discussed in detail in the analyses of the AC susceptibility measurements. The latter are a useful way to identify the freezing dynamics of the spin-glass (SG) materials.

3.5. AC Susceptibility

For measurements of AC susceptibility (${\chi }_{\mathrm{ac}}$), an AC magnetic field ($H_{\mathrm{ac}}$) is applied to the sample and, as a consequence, a resultant magnetic moment is measured. The ${\chi }_{\mathrm{ac}}$ is represented as follows:

$${\chi }_{\mathrm{ac}}={\chi }^{\prime }+i{\chi }^{″}$$

where χ’ real and ${\chi }^{″}$ imaginary parts are, respectively, the in-phase and out-phase components of ${\chi }_{\mathrm{ac}}$. It should be noted that the relaxation time ($\tau$) of the AC susceptibility measurement is not based upon the energy barrier ($E_{\mathrm{a}}=K_{\mathrm{eff}}V$ where $K_{\mathrm{eff}}$ is the effective anisotropy constant and $V$ is the volume of particles). However, it is influenced by the external excitation frequency. The AC susceptibility measurements give important details about the dynamics of the systems and the strength of exchange interactions between the magnetic nano-particles (MNPs) and between the different cations.

Firstly, we will discuss the χ’ real part measurements of the two NPs samples with x = 0.00 and 0.03. Figure 6 presents the curves of χ’versus T, ranging from 350 to 2 K, for x = 0.00 and 0.03 products, performed in the presence of an $H_{\mathrm{ac}}$ = 10 Oe and in a frequency range of 50–10⁴ Hz. The magnitude of χ’ for x = 0.03 increased slightly compared to the non-substituted product (x = 0.00), which is in accordance with M_ZFC (T) and M_FC (T) measurements. The in-phase AC susceptibility data of the different samples showed dispersion and a decrease in magnitude while increasing the applied frequency from 50 Hz to 10 kHz. The χ’(T) curve of the non-substituted NiFe₂O₄ NPs exhibited a peak at around 300 K. However, the x = 0.03 product did not show any peak up to 350 K.

Figure 7 shows the ${\chi }^{″}\left(T\right)$ curves for NiFe_2-xDy_xO₄ (where x = 0.00 and 0.03) performed in a $H_{\mathrm{ac}}$ = 10 Oe and in the frequency range of 50–10⁴ Hz. It can be seen that both samples display two peaks—the first at the higher temperature indicated by T_B in the figure, which is associated with magnetic blocking of huge core spins, and the second indicated by $T_{\mathrm{s}}$, which can be associated with the spin-glass freezing on the surface of a single NP [28]. The χ’(T) curves do not offer any information about these two peaks, hence from now on we will focus only on analyses of ${\chi }^{″}\left(T\right)$ curves.

At the same applied frequency, the T_B and $T_{\mathrm{s}}$ shifted to lower temperatures with Dy substitution compared to the non-substituted one. This is consistent with the M_ZFC analyses. Both the blocking temperature T_B and spin-glass freezing temperature $T_{\mathrm{s}}$ are affected by frequency. Both show a shift to higher temperatures upon increasing the value of the applied frequency ($f$). Similar behavior is observed in the spin-frustrated system of CoFe₂O₄ NPs dispersed in an SiO₂ matrix [28]. The shifting with f is helpful for evaluating dynamic magnetic behaviors, deducing the anisotropic energy, the magnetic anisotropy, and the interaction strength between MNPs.

Various physical laws can be used to investigate the f-dependence shift of T_B and $T_{\mathrm{s}}$ temperatures. The Neel–Arrhenius (N–A) law was first tested to fit the experimental data (Figure 8). This theory is valid for thermal excitations of non-interacting single-barrier NPs and is expressed as follows [29,30]:

$$\tau ={\tau }_{\mathrm{o}} exp\left(E_{\mathrm{a}}/k_{\mathrm{B}}T\right)$$

where $\tau =1/f$ is the measured time, ${\tau }_0$ is the jump attempt time (in the range of 10⁻⁹–10⁻¹³ s), $k_{\mathrm{B}}$ is the Boltzmann constant, and $E_{\mathrm{a}}=K_{\mathrm{eff}}V$ is the activation energy barrier. The estimated values of ${\tau }_0=1/f_0$, $E_{\mathrm{a}}/k_{\mathrm{B}}$ and $K_{\mathrm{eff}}$ for different samples are given in

Table 3. Fitting parameters (${\tau }_{\mathrm{o}}$, $E_0/k_{\mathrm{B}}$ and ) of prepared NiFe_2−xDy_xO₄ (x = 0.00 and 0.03) NPs estimated using different laws.

Models	Parameters	Values
		Peak (TB)		Peak (TS)
		x = 0.00	x = 0.03	x = 0.00	x = 0.03
Neel–Arrhenius	${\tau }_o$ (s)	1.05 × 10−70	1.68 × 10−32	1.69 × 10−16	1.26 × 10−392
	$E_0/k_{\mathrm{B}}$ (K)	49,083	17,403	2150	184,793
	$K_{\mathrm{eff}}$ (erg/cm3)	3.09 × 105	1.09 × 105	1.35 × 104	1.16 × 106
Vogel–Fulcher	${\tau }_{\mathrm{o}}$ (s)	5.45 × 10−10	1.32 × 10−9	8.00 × 10−7	6.06 × 10−46
	$E_0/k_{\mathrm{B}}$ (K)	367.57	621.53	100.38	2188.95
	$T_{\mathrm{o}}$ (K)	292.71	216.76	58.82	183.84
	$K_{\mathrm{eff}}$ (erg/cm3)	2.32 × 103	3.92 × 103	6.34 × 102	1.38 × 104
Critical slowing down	${\tau }_{\mathrm{o}}$ (s)	****	****	2.88 × 10−9	2.50 × 10−12
	$T_{\mathrm{g}}$ (K)	****	****	58.39	205.55
	$z\upsilon$	****	****	5.77	3.95

. The best N–A fit offers very unreasonable values for ${\tau }_0$ and $E_{\mathrm{a}}/k_{\mathrm{B}}$. This indicates that the synthesized products do not obey the thermally activated N–A law and, as a consequence, they are non-interacting.

The Vogel–Fulcher (V–F) law is a useful model for investigating the interactions between NPs. This law uses an additional parameter, $T_0$, that represents the strength of inter-particle interactions. Based on this model, the relaxation is described as follows [29,30]:

$$\tau ={\tau }_{\mathrm{o}} \exp \left[E_{\mathrm{a}}/k_{\mathrm{B}}(T-T_0\right)].$$

The fitting data using the V–F law of the plots of $f$ vs. T_B and $f$ vs. $T_{\mathrm{s}}$ for the prepared products are illustrated in Figure 9a,b, respectively. The different estimated parameters are summarized in Table 3. The analysis of f-dependent T_B now gives reasonable ${\tau }_0$ and $E_{\mathrm{a}}/k_{\mathrm{B}}$ values. Obviously, the T₀ values are not negligible compared to T_B. The occurrence of T₀ confirms the presence of moderate inter-particle interactions between the NPs [28,29,30]. It is found, moreover, that ${\tau }_0$ increased more for the x = 0.03 product than for the x = 0.00 one. The increase in the ${\tau }_0$ for x = 0.03 product suggests the strengthening of interactions between NPs [29,30]. Compared to the x = 0.00 product, the values of $E_{\mathrm{a}}/k_{\mathrm{B}}$ and $K_{\mathrm{eff}}$ improved with Dy substitution for x = 0.03. This improvement iresulted from the strengthening of magnetic interactions among different NPs and the increase of magnetic anisotropy sources [29,30].

In other hand, the investigation of f vs. $T_{\mathrm{s}}$ provides unphysical values for ${\tau }_0$. Therefore, the critical slowing down (CSD) law is used to study the presence of SG behavior in the synthesized NPs. Based on this model, the relaxation is expressed as [30]:

$$\tau ={\tau }_0^{*}{\left[\frac{T_{\mathrm{s}}}{T_{\mathrm{g}}}-1\right]}^{-\left(z\upsilon \right)}$$

where ${\tau }_0^{*}$ is associated to the coherence time of coupled individual “atomic” spins in the NP (in the range 10⁻⁶–10⁻¹³ s) [31], $T_{\mathrm{g}}$ is the SG freezing temperature, and $T_{\mathrm{s}}$ is the f-dependent freezing temperature. The $“z\upsilon ”$ is the critical exponent that offers information about the SG, and it varies from 4 to 12 for various SG systems [28]. We fit the same f vs. $T_{\mathrm{s}}$ data using the CSD law, in order to examine the possibility of the SG nature (Figure 10). The various deduced parameters are listed in Table 3. The obtained reasonable values of ${\tau }_0$, $T_{\mathrm{g}}$ and $“z\upsilon ”$ proved the existence of SG behavior in the prepared samples. Similar comportment has been reported in numerous products, such as CoFe₂O₄/(SiO₂)_x systems [28], Fe₃O₄ MNPs ($z\upsilon =8.2$ and $\tau ~ {10}^{-9} s$) [32,33], soft ferrite Ni_0.3Zn_0.7Fe₂O₄ NPs ($z\upsilon =8.01$ and $\tau ~ {10}^{-12} s$) [34], and La_0.9Sr_0.1MnO₃ NPs [35]. It is reported that the strength of magnetic interactions increases based on the decreasing $“z\upsilon ”$ exponent. The non-substituted NiFe₂O₄ product exhibits a $“z\upsilon ”$ value equal to 5.11, and it decreases to 3.95 with Dy substitution for x = 0.03. This result indicates the improvement of the magnetic interactions among NPs for the x = 0.03 product.

4. Conclusions

Series of Dy³⁺ doped ferrite nanoparticles of NiFe_2-xDy_xO₄ (0.0 ≤ x ≤ 0.1) NPs were prepared using a hydrothermal method. The as-prepared specimens were thoroughly characterized using various analytical measurement techniques to determine their structure, morphology, and distinct magnetic traits. An XRD pattern of nanoferrites confirmed the formation of a spinel cubic structure. The Mössbauer spectra displayed the cation distribution, verifying the occupation of Dy³⁺ ions at octahedral B sites. The measurements of ZFC-FC magnetization and AC susceptibility were examined. The magnetic data showed the existence of two critical temperatures. The first one is the blocking temperature $T_{\mathrm{B}}$, which corresponds to the magnetic blocking of huge core spins, and the second is $T_{\mathrm{s}}$, which can be associated to spin-glass freezing arising on the surface of a single NP. The $T_{\mathrm{B}}$ and $T_{\mathrm{s}}$ temperatures vary with the Dy substitution. This is due to numerous intrinsic and extrinsic factors that principally include magneto-crystalline factors, interactions among particles, surface, and shape anisotropy. Furthermore, both the $T_{\mathrm{B}}$ and $T_{\mathrm{s}}$ temperatures are affected by applied frequency. Various physical laws, such as Neel-Arrhenius, Vogel-Fulcher, and critical slowing down, are used to investigate the f-dependence shift of the $T_{\mathrm{B}}$ and $T_{\mathrm{s}}$ temperatures. It was found that the Dy substitution enhances the magnetic interactions. Compared to the x = 0.00 product, the $E_{\mathrm{a}}/k_{\mathrm{B}}$ and $K_{\mathrm{eff}}$ values improved for x = 0.03. This proved that the magnetic interactions are enhanced due to the Dy substitution.